Lesson 7: Slopes and Functions: Speed and Velocity 7.1 Observe and Represent Another way of comparing trend lines is by calculating the slope of each line and comparing the numerical values of the slopes. a) Use the graph above and calculate the slope of the line for each case. Explain how you calculated the slope. Need Some Help? Slope: Often used to describe the measurement of the steepness of a straight line. A higher slope value indicates a steeper incline. The slope is defined as the ratio of the change in the value of the dependent variable (vertical change) over the change in the value of the independent variable (horizontal change). In other words, vertical change divided by horizontal change! b) For the skiers, what do you think the slope of the line represents? Try to answer using your common sense. c) What are the units of slope? How do you know? d) Refer to the graphs to check if your answer makes sense. How do you know? Is there anything else you notice? Explain. 34 PUM Kinematics Lesson 7: Slopes and Functions: Speed and Velocity Copyright 2009, Rutgers, The State University of New Jersey.
7.2 Represent and Reason a) Recall the lab you did in activity 6.3. Write a function (expression) that will allow you to find position (x) for any time (t) for the first ball. Then write a second expression for the second ball. Need Some Help? When mathematicians and physicists express patterns mathematically they use functions. A function is a rule that one uses to find a dependent variable when an independent variable is known. You may have met functions in a math class. There the independent variable was labeled x and the dependent variable is labeled y. The function then is y(x). In science and math class you can actually use any labels as long as you agree on which was the independent and which is the dependent variable. For the problem below, the independent is t, and a dependent is x. Example: Examine: Define: Represent: Describe the relationship between the two variables. Describe the variables used in the scenario Write a mathematical equation using variables The object changes its position by 50 meters each second t = time elapsed x = position x = 50t Time (second) Position (meters) 1 50 2 100 3 150 This expression can all be written in function notion as x(t) = 50t, however, in physics it is necessary to include units of measure x(t) = 50(m/s) t or x(t)=50(m/s) * t. x(t) is read as x of t. REMEMBER! When you graph a function the independent variable is always placed on the horizontal and the dependent variable on the vertical axis. b) Use the function provided to complete the tables: Examine: Describe the relationship between the two variables. Table of Values Define: Represent: Describe the variables used in the scenario Write a mathematical equation using variables x(t) = ¼ (m/s) * t PUM Kinematics Lesson 7: Slopes and Functions: Speed and Velocity Copyright 2008, Rutgers, The State University of New Jersey. 35
7.3 Represent and Reason a) Imagine a man started his journey at 3 pm and ended at 3.30 pm. At 3 pm he was at a supermarket and at 3.30 pm he was at position 2.5 km away from the supermarket. How fast was he travelling? Explain how you got your answer. Express it in different units. Did You Know? Variable and notation are very important in science and math. Specifically in physics, we must be clear about what each variable, notation, and expression means. Example: A woman started her journey at time t 1 and ended at time t 2. At time t 1, she was at position, x(t 1 ) = x 1 and at time t 2, she was at position x(t 2 ) = x 2. As you can see, this notation is very overwhelming! This is why we omit x(t 1 ) and x(t 2 ) and simplify it as x 1 and x 2. However, it is necessary to know that every time you see x with a subscript, the subscript indicates a position at a specific clock reading (time). Use the variables (no numbers) from the information box above to answer each question: b) When did the woman start her trip? c) How far did the woman travel? d) How fast did the woman travel? Here s an Idea! The format that you wrote in part (d) is typically how physicists define velocity. 7.4 Hypothesize Write a function (x)t for a man walking at a speed of 0.3 m/s who starts moving from position zero at a zero clock reading. 36 PUM Kinematics Lesson 7: Slopes and Functions: Speed and Velocity Copyright 2009, Rutgers, The State University of New Jersey.
7.5 Test Your Idea If these functions that you have been writing really do represent motion accurately, it should be possible to use them to predict position of a moving object if we know the type of motion that occurs. To do so we will use the PhET simulation, The Moving Man. You should navigate to the following web address and click, Run Now. http://phet.colorado.edu/simulations/sims.php?sim=the_moving_man a) Use the function for the man from 7.4 to predict where the man will be after 8 second b) In the Moving Man simulation, type 0.3 in the velocity box. Make sure that the man begin at position zero and zero clock reading. c) Then click Go. Just before the man gets to the wall click, Stop. This can be located at the bottom of the simulation. d) Move the gray bar on the graph to 8 seconds to check your mathematical prediction of the position. e) Did your prediction match the actual outcome? Make sure to check you work and discuss any problems you may have had. You can use the space below. 7.6 Represent and Reason a) Determine the slope of the trend line shown above using data from the graph. Would the slope change if you used different data points? Use two more examples to determine the slope. b) Write an equation for the function that will allow you to find a value for x for any value of t. c) Sketch a graph that represents the motion of an object traveling at the same speed but in the opposite direction. PUM Kinematics Lesson 7: Slopes and Functions: Speed and Velocity Copyright 2008, Rutgers, The State University of New Jersey. 37
d) Calculate the slope for the graph and write an equation for the function that will allow you to find x for any t. e) What is the same about the functions you wrote in part (e) and part (f). How are they similar? How are they different? 7.7 Represent and Reason a) A train is travelling at 22 m/s when it passes your town. Where will the train be 30 minutes later? Write a function (x)t for the train. b) Imagine that the train was 300 meters south of the your town when it was first observed. (We will assume that south is a negative direction) How would the function change? Write a new function for x(t). Need Some Help? The method we have been using to write functions is great, but it only works when an object s motion begins at the origin. If this is not the case, the starting position of the object must be included in our function! Consider the problem from earlier in the lesson. Before the object started at 0 and our function was x(t) = 50(m/s)t. Now that the object is starting at 1 meter from the origin, how could we change our function to fit the data? Time (sec) Position (m) 0 1 1 51 2 101 3 151 Hint: What could you add at the end of the function? c) Imagine that instead of motion, you are studying the amount of money in your pocket. Amount of money is the dependent variable and time is the independent variable. You start with 10 dollars and then spend 2 dollars every hour for 5 hours. Sketch a graph of number of dollars in your pocket versus time, then write a function representing the amount of money in your pocket as a function of time. Did You Know? 38 PUM Kinematics Lesson 7: Slopes and Functions: Speed and Velocity Copyright 2009, Rutgers, The State University of New Jersey.
Position, distance, displacement and path length: These refer to different things! Position x is the location of an object relative to a chosen zero on the coordinate axis. Displacement x 2 - x 1 indicates a change in position and has a sign indicating the direction of the displacement. The magnitude of that position change is the distance and is always positive. Path length refers to the total length of the path that was travelled. Velocity and speed for constant speed linear motion: Velocity is the slope of the position versus clock reading graph or the ratio of the displacement of an object during a time interval divided by that time interval. The unit of velocity is m/s, miles/h, km/h, and so forth. Positive velocity means that the object is moving in the positive direction, negative means in the negative direction. Speed is the magnitude of velocity and is always positive. For constant speed, motion in a straight line, velocity can be written as: x 2 x 1 = (Δx) = change in position t 2 t 1 = (Δt) = time interval 7.8 Observe and explain Homework PUM Kinematics Lesson 7: Slopes and Functions: Speed and Velocity Copyright 2008, Rutgers, The State University of New Jersey. 39
The motion of two rockets is represented graphically. a) Draw a picture for the motion of the two rockets; b) Create a dot diagram for each of the rockets; c) Calculate the slope of each line. What is the speed of each rocket? d) Write a mathematical expression x(t) for lines A and B. e) How much distance was traveled by rocket A after 6 minutes? f) How much time did it take rocket B to travel 60 km? g) How much sooner did rocket B travel 60 km than rocket A? 7.9 Reason A motorcyclist rides west at 40 m/s on a straight highway. The illustration below represents the car s position at a zero clock reading using three different reference frames. Use this to answer the following questions. v 200 100 0 100 200 x (m) 100 0 100 200 300 x (m) 0 100 200 300 400 x (m) 40 PUM Kinematics Lesson 7: Slopes and Functions: Speed and Velocity Copyright 2009, Rutgers, The State University of New Jersey.
a. How will the velocity in the second reference frame differ from the first and third reference frames? Explain your reasoning. b. Write a function in for each reference frame. c. Would a dot diagram differ between the different reference frames? If so, explain how. d. Would a position-versus-time graph differ for the difference reference frames? If so, explain how. PUM Kinematics Lesson 7: Slopes and Functions: Speed and Velocity Copyright 2008, Rutgers, The State University of New Jersey. 41